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Stability of the rarefaction wave in the singular limit of a sharp interface problem for the compressible Navier-Stokes/Allen-Cahn system

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Abstract

This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space. For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation, and where the strength of the initial phase field could be arbitrarily large, we prove that the solution of the Cauchy problem exists for all time, and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero. The proof is mainly based on a scaling argument and a basic energy method.

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Anshun University, Anshun, 561000, China

    Yunkun Chen

  2. College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, 100029, China

    Bin Huang & Xiaoding Shi

Authors
  1. Yunkun Chen
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  2. Bin Huang
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  3. Xiaoding Shi
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Correspondence to Xiaoding Shi.

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Conflict of Interest The authors declare that they have no conflict of interest.

Additional information

Chen’s work was supported by the National Natural Science Foundation of China (12361044); Shi’s work was supported by the National Natural Science Foundation of China (12171024, 11971217, 11971020). This paper was also supported by the Academic and Technical Leaders Training Plan of Jiangxi Province (20212BCJ23027).

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Chen, Y., Huang, B. & Shi, X. Stability of the rarefaction wave in the singular limit of a sharp interface problem for the compressible Navier-Stokes/Allen-Cahn system. Acta Math Sci 44, 1507–1523 (2024). https://doi.org/10.1007/s10473-024-0417-3

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  • DOI: https://doi.org/10.1007/s10473-024-0417-3

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